For a 0.001 ETH transaction, about how much fee would be deducted?
The fee you pay for a transaction doesn't depend on the transaction amount at all.
The current fee for a ethereum transaction is around $0.05 worth of ethereum. If you send tokens like USDT, the fee would be higher. For a USDT transaction, the current fee is around $0.1 worth of ethereum.
Note that the fees can change over time and custodial services may charge you higher fees for your withdrawals.
As I already said, I made a USDT withdrawal from bet105 a few days ago and they didn't charge me any fee for that.
It depends on many factors, but if they are covering the fees - that's great.
I heard that ARB network is good for the fees, but they've got only ERC20 as of now.